Some capacity-theoretic results extended to algebraic curves
Walters, Nathan Lawrence
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This work deals with generalizing two classical theorems of capacity theory; a Ferguson-type result on approximation by rational functions with prescribed polar divisor and integrality conditions, and a Polya-type criterion for when analytically defined objects have a natural algebraic structure. These results are expanded in two main directions. The primary goal is to extend the results to algebraic curves. The secondary goal is to establish the results over all global fields, and not just number fields. Both goals are accomplished; the main tool for the primary goal is a generalization of the notion of capacity formalized by Robert Rumely; the tools used for the secondary goal are mostly ad-hoc arguments for the specific problems encountered in the generalizations.